Factors Affecting Conductivity

Electron Mobility is Decreased by Defects

Increasing the temperature of a conductor decreases its conductivity, because the resulting lattice vibrations disrupt the resonant/fractional bonds, which makes it harder electrons to move between them. Thermal disruption of the electronic structure of a material is constantly fluctuating, but any defects in the crystal will have a similar effect. All defects change the electronic bonding in their vicinity which reduces the mobility of electrons. This includes:

• Point defects (impurity atoms, vacancies, and interstitial atom)
• 1D defects (i.e., dislocations)
• 2D defects (i.e., grain boundaries)

In the free electron model, all of these defects increase the probability of an electron scattering, which slows it down. In this model, the velocity of an electron looks like Figure 13.8.1 below, and we call the average velocity the drift velocity, $v_d$. In between collisions, velocity increases linearly due to the force of the electric field, $\mathcal{E}$. So, $v_d \propto \mathcal{E}$. We call the proportionality constant between drift velocity and electric field, the electron mobility, $\mu_e$. So:

$$v_d = \mu_e \mathcal{E}$$

And the mobility is related to the frequency of scattering events, $f_s$. The more scattering events there are, the lower the drift velocity will be: $$\mu_e \propto \frac{1}{f_s}$$.

The three factors affecting conductivity

Overall, the conductivity of a material depends on three factors, expressed in the following equation:

Where:

• $n$ is the concentration of free electrons ($/ \text{m}^3)$. This was explored in the first question of Exercise 13.4.1
• $|e|$ is the magnitude of electron charge ($1.6 \times 1-^{-19} \mathrm{C})$
• $\mu_3$ is the electron mobility discussed above ($\mathrm{m}^{2} \mathrm{V}^{-1} \mathrm{s}^{-1}$)

Remember that conductivity is an intrinsic property of a material, independent of its shape. To calculate what the actual current through a wire will be be due to an applied voltage, you need to take into account the length and cross sectional area.

Matthiessen's Rule

Matthiessen's Rule states that the overall resistivity, $\rho$, of a material (recall that resistivity is just the inverse of conductivity: $\rho = \frac{1}{\sigma}$) can be approximated by adding up the contributions of the different factors discussed above:

$$\rho_{\mathrm{total}} = \rho_{\mathrm{deformation}, d} + \rho_{\mathrm{impurity}, i} + \rho_{\mathrm{thermal}, t}$$

These three factors can all be investigated empirically:

• $\rho_{\mathrm{deformation}, d}$ relates to the amount of cold working which introduces dislocations into the material.
• $\rho_{\mathrm{impurity}, i}$ relates to the weight percent of impurity atoms in the lattice and
• $\rho_{\mathrm{thermal}, t}$ relates to the temperature.

Figure 13.8.2 below shows the resistivity vs temperature for copper with different amounts of the other two contributions to the resistivity.